Characterizing a sample by material basis decomposition

ABSTRACT

A method for characterizing a sample, by estimating a plurality of characteristic thicknesses, each being associated with a calibration material, including acquiring an energy spectrum (S ech ) transmitted through this sample, located in an X and/or gamma spectral band, naled spectrum transmitted through the sample; for each spectrum of a plurality of calibration spectra (S base (L k ; L l )), calculating a likelihood from said calibration spectrum (S base (L k ; L l )), and from the spectrum transmitted through the sample (S ech ), each calibration spectrum (S base (L k ; L l )) corresponding to the energy spectrum transmitted through a stack of gauge blocks, each formed of a known thickness of a calibration material; estimating the characteristic thicknesses (L 1 , L 2 ) associated with the sample according to the criterion of maximum likelihood.

TECHNICAL FIELD

The present invention relates to the field of characterizing a sample byX spectrometry (wavelength lower than 10⁻⁸ m) and/or gamma spectrometry(wavelength lower than 10⁻¹¹ m). The invention more particularly relatesto determining a series of characteristic thicknesses, each beingassociated with a calibration material. A material basis decompositionis usually mentioned.

STATE OF PRIOR ART

In prior art, different solutions are known to characterize a sample, inparticular by X and/or gamma spectrometry.

It could be shown in particular that for samples having a quite low meaneffective atomic number, the attenuation function of the sample L*μ(E)can be theoretically described as a linear combination of the respectiveattenuation coefficients μ_(MAT1), μ_(MAT2) of two calibration materialsMAT1 and MAT2. In other words, the sample of a thickness L generates thesame attenuation as a thickness L₁ ^(ech) of the material MAT1 plus athickness L₂ ^(ech) of the material MAT2:

μ_(MAT1)(E)*L ₁ ^(ech)+μ_(MAT2)(E)*L ₂ ^(ech) =L*μ(E)   (1).

Materials MAT1 and MAT2 are named calibration materials, since theirrespective attenuation coefficients define a decomposition basis tocharacterize the sample. Thicknesses L₁ ^(ech) and L₂ ^(ech) are said tobe characteristic, since they characterize the sample in thedecomposition basis. The attenuation function is the product of athickness of a material by its attenuation coefficient. Each attenuationcoefficient is a function of the energy. The attenuation function isconsidered over a plurality of energy channels (also referred to asenergy bands or energy gaps).

FIG. 1 illustrates this property. It is a graph depicting on theabscissa the energy in kilo-electron-volt (keV) according to alogarithmic scale, and on the ordinate the value of the attenuationfunction according to a logarithmic scale (without unit). Curve 101depicts the attenuation function μ_(ECH)(E)*L of a 10 mm thickpolytetrafluoroethylene sample. Curve 102 depicts the attenuationfunction μ_(MAT1)(E)*L₁ ^(ech) of a 1.49 mm thick graphite sample. Curve103 depicts the attenuation function μ_(MAT2)(E)*L₂ ^(ech) of a 7.66 mmthick aluminium sample. The sum of curves 102 and 103 corresponds tocurve 101, so that the sample can be defined by L₁ ^(ech)=1.49 mm and L₂^(ech)=7.66 mm, the calibration materials being graphite and aluminium.The different attenuation functions are referred to as linear, sincethey have no discontinuity (k-edge).

In practice, the attenuation function of the sample is measured by wayof the device 200 such as depicted in FIG. 2. The device 200 comprises asource of electromagnetic radiation 201, emitting an analysis beam 209in the X and/or gamma spectral band. The device 200 also comprises adetector 203 able to count a number of received photons, for each energychannel of a plurality. The source 201 and the detector 203 formtogether a spectrometer. The sample 202, of a thickness L, is providedbetween the source 201 and the detector 203, and is crossed by theanalysis beam 209. The analysis beam 209 is attenuated by crossing thesample according to the Beer-Lambert law:

S _(ECH)(E)=S ₀(E)exp(−μ_(ECH) *L)   (2)

S₀(E) is the energy spectrum measured in the absence of the sample, andS_(ECH)(E) is the energy spectrum measured in the presence of thesample, named spectrum transmitted through the sample.

The attenuation function of the sample is then defined by:

$\begin{matrix}{{\mu_{ECH}*L} = {- {\ln \left( \frac{S_{ECH}(E)}{S_{0}(E)} \right)}}} & (3)\end{matrix}$

Theoretically, the attenuation function of the sample, then theattenuation function of a thickness L₁ of the calibration material MAT1(named first attenuation function) and the attenuation function of athickness L₂ of the calibration material MAT2 (named second attenuationfunction) are measured. Then, the coefficients a and b assigned to thefirst and second attenuation functions are searched for to define theattenuation function of the sample as a linear combination of the firstand second attenuation functions. L₁ ^(ech) and L₂ ^(ech), or moreprecisely the estimations of these lengths, {circumflex over (L)}₁^(ech) and {circumflex over (L)}₂ ^(ech), are deduced therefrom.

In practice, the measurement of the energy spectra is prone to errors.These errors especially come from the photon noise (statisticalfluctuation of the number of photons which interact in thespectrometer), from the width of the energy channels detected by thespectrometer, from the electronic noise and other imperfections of thespectrometer. These errors recur in the measured attenuation function ofthe sample, and in the first and second attenuation functions. Theseerrors affect the linearity of the relationship expressed in theequation (1), and prevent the characteristic thicknesses L₁ ^(ech) andL₂ ^(ech) from being estimated as described above.

A known solution consists in modelling the detection chain by a responsefunction of the system, to get rid of the errors brought by the formerby inverting the function of the system. A drawback of this solution isthat it is dependent on the quality of said modelling, a high qualitymodelling being difficult to achieve.

U.S. Pat. No. 8,929,508 provides an analytical formula directlysupplying an estimation of the characteristic thicknesses L₁ ^(ech) andL₂ ^(ech), based on the hypothesis that the relationship (1) is linearon each of a plurality of small gaps of thicknesses and of transmittedspectra. Such a hypothesis however does not provide a sufficientlyaccurate estimation of the characteristic thicknesses, especially as itrests on the hypothesis according to which the attenuation by the samplein each energy channel follows a Gaussian distribution, whereas aPoisson distribution is the most realistic hypothesis.

An object of the present invention is to provide a method and a deviceallowing an accurate estimation of the characteristic thicknesses usedto characterize a sample in a calibration material basis.

DISCLOSURE OF THE INVENTION

This object is achieved with a method for characterizing a sample, byestimating a plurality of thicknesses, named characteristic thicknesses,each being associated with a material, named calibration material. Themethod according to the invention comprises the following steps:

-   -   acquiring an energy spectrum, named spectrum transmitted through        the sample, said spectrum being defined by a number of photons        transmitted through the sample in each channel of a plurality of        energy channels located in an X and/or gamma spectral band;    -   for each spectrum of a plurality of energy spectra, named        calibration spectra, calculating the value of a likelihood        function from said calibration spectrum and from the spectrum        transmitted through the sample, each calibration spectrum        corresponding to the spectrum transmitted through a stack of        gauge blocks, each gauge block being formed of a known thickness        of a calibration material;    -   determining the estimations of the characteristic thicknesses        associated with the sample, from said values of a likelihood        function and according to the criterion of maximum likelihood.

Thus, the limitations associated with the measurement errors existing inprior art described in the introduction are overcome.

The characteristic thicknesses of calibration materials, characterizinga sample in a calibration basis, can be estimated directly from measuredspectra and without resorting to purely theoretical spectra obtained bymodelling an acquisition chain. Thus, the inaccuracies related to theimperfections of such a modelling are overcome.

The estimation of the characteristic thicknesses is not based on anyhypothesis which is particularly simplifying.

The measurement error is smaller, by way of using the series ofcalibration spectra forming a calibration material decomposition basis,named calibration basis.

Furthermore, as will be detailed later, such a method enables the rightmodel of a behaviour of the attenuation in each energy channel to beused (behaviour which can be referred to as measurement noise, andmodelled by a Poisson statistical distribution). The use of the rightestmodel provides an accurate estimation of the characteristic thicknesses.

The invention also relates to a computer program product arranged toimplement a method according to the invention.

Finally, the invention relates to a device for characterizing a sample,comprising an electromagnetic source emitting into an X and/or gammaspectral band, and a detector for measuring a spectrum transmittedthrough the sample, said spectrum being defined by a number of photonstransmitted through the sample in each channel of a plurality of energychannels. According to the invention the device comprises a processor,arranged to implement the method according to the invention, and amemory receiving the calibration spectra, the memory being connected tothe processor.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be better understood upon reading the description ofexemplary embodiments given by way of purely indicating and in no waylimiting examples, with reference to the accompanying drawings in which:

FIG. 1 illustrates the linear attenuation functions of a sample and oftwo calibration materials, used in the methods according to prior art;

FIG. 2 schematically illustrates a device for measuring an energyspectrum according to the invention;

FIG. 3 schematically illustrates a first embodiment of the methodaccording to the invention;

FIG. 4 schematically illustrates a device according to the invention;

FIG. 5 illustrates calibration spectra according to the invention;

FIG. 6 schematically illustrates a second embodiment of the methodaccording to the invention; and

FIG. 7 illustrates an alternative for the embodiment of FIG. 6; and

FIG. 8 illustrates a calibration basis enhanced according to theinvention.

DETAILED DISCLOSURE OF PARTICULAR EMBODIMENTS

The invention relates to a method for characterizing a sample.Characterizing here consists in defining a spectrum of the energytransmitted through the sample of a thickness L, as being the spectrumof the energy transmitted through a stack of two gauge blocks or more,each consisting of a different calibration material.

Each gauge block of a calibration material has a thickness, namedcharacteristic thickness, or equivalent thickness, which is attempted tobe estimated and which characterizes the sample.

The calibration materials are materials the respective attenuationcoefficients of which define a decomposition basis to characterize anysample. Two calibration materials or even more are considered.

Thus, a linear combination of the spectra transmitted through arespective calibration material, each associated with a characteristicthickness, can be associated with the sample.

The aim is, in an equivalent way, to associate to the sample a linearcombination of the attenuation coefficients of calibration materials,each associated with a characteristic thickness (the energy spectrumS₀(E) in the absence of a sample and a gauge block being a constant).

The method according to the invention is implemented in a device 400such as depicted in FIG. 4. The source 401 and the detector 403correspond to the source 201 and the detector 203 described withreference to FIG. 2.

The thickness corresponds to the distance travelled in the sample 402,respectively in a gauge block of a calibration material by an analysisbeam 409 emitted by the electromagnetic radiation source 401 andreceived by the detector 403 after crossing the sample 402, respectivelythe gauge block (refer to FIG. 4).

The sample here refers to any object, in particular a biological sample,such as a biological tissue. The method according to the invention doesnot dictate any condition regarding its mean effective atomic number.Nevertheless, the latter is advantageously lower than 30.

During an initial calibration step, a series of spectra is acquired,each being a spectrum of the energy transmitted through a stack of gaugeblocks. Each energy spectrum is defined by a number of photonstransmitted through the stack, in each channel of a plurality of energychannels located within an X and/or gamma spectral band. Such an energyspectrum is referred to as a calibration spectrum.

Each gauge block, also referred to as a standard, or a calibrationsample, consists in a calibration material, and has a known thickness.It is for example a lamella consisting of said material. The pluralgauge blocks of the stack are formed of distinct calibration materials.

This step is preferably implemented only once, the same calibrationspectra being used afterwards to characterize any sample.

This step is preferably implemented in the device of FIG. 4, in the samemeasurement conditions as the spectrum transmitted through the samplesuch as hereafter defined (same spectrometer, same emission power, samedistance between the source and the detector, same irradiationduration). When necessary, several measurements on a same stack of gaugeblocks are averaged in order to circumvent the photon noise.Alternatively, the calibration spectra are calculated from a numericalmodelling of the device of FIG. 4, in particular a numerical modellingof the acquisition chain.

In a first step 31, the spectrum transmitted through the sample,S^(ech), is acquired using the detector 403. The spectrum transmittedthrough the sample is the spectrum of the energy transmitted throughsaid sample. It is defined by a number of photons per energy channel,for each channel of a plurality of energy channels. The spectrumtransmitted through the sample comprises at least two energy channels,preferably several tens. For example, the energy channels cover togetherall the energy band ranging from 10 to 120 keV, and each have a width of1 keV. The energy channels of the spectrum transmitted through thesample are preferably the same as those of the calibration spectra.

For each of the plurality of calibration spectra, in step 32, the valueof a likelihood function is calculated, corresponding to the likelihoodof the calibration spectrum taking into account the spectrum transmittedthrough the sample S^(ech). The different energy channels of a samespectrum are processed together, and not independently from each other.

Each calibration spectrum being associated with a known thickness of acalibration material, the likelihood function according to the inventiondepends on the thicknesses of the calibration materials.

Two calibration materials MAT1 and MAT2 are for example considered. Acalibration spectrum S^(base)(L₁; L₂) is the energy spectrum transmittedthrough a gauge block of a thickness L₁ of a material MAT1 joined to agauge block of a thickness L₂ of a material MAT2. The likelihoodfunction calculated from the spectrum transmitted through the sampleS^(ech) and from the calibration spectrum S^(base)(L₁; L₂) thus dependson L₁ and on L₂.

FIG. 5 illustrates a series of calibration spectra. L₁ can take 4 values0, 1, 2 or 3 (length unit). L₂ can take 4 values 0, 1, 2 or 3 (lengthunit). 4×4=16 calibration spectra are thus obtained, corresponding toall the possible combinations of L₁ and L₂.

Let L₁ be able to take N1 values, and L₂ be able to take N2 values, atthe end of step 32 N1×N2 values of the likelihood function are obtained.

In step 33, these values of the likelihood function are used todetermine an estimation of a thickness L₁ ^(ech) of the material MAT1and of a thickness L₂ ^(ech) of the material MAT2, such that thespectrum transmitted through the sample corresponds to the spectrumtransmitted through the juxtaposition of a gauge block of a thickness L₁^(ech) of a material MAT1 and of a gauge block of a thickness L₂ ^(ech)of a material MAT2. The estimated values of L₁ ^(ech) and L₂ ^(ech) arenoted {circumflex over (L)}₁ ^(ech) and {circumflex over (L)}₂ ^(ech).

The aim is to search for, by means of the calibration basis, thecalibration spectrum consisting of a thickness L₁ of the material MAT1and of a thickness L₂ of the material MAT2, which most resembles thespectrum transmitted through the sample.

The criterion used for the estimation is the maximum likelihood.

In the example illustrated in FIG. 3, it is searched the maximum valueamong the N1×N2 values of the previously calculated likelihood function.In other words, in the calibration basis, the calibration spectrum whichmost resembles the spectrum transmitted through the sample is searchedfor. The thicknesses L₁, L₂ associated with this maximum value of thelikelihood function correspond to the estimations {circumflex over (L)}₁^(ech) and {circumflex over (L)}₂ ^(ech) of the characteristicthicknesses L₁ ^(ech) and L₂ ^(ech).

For example, let a thickness L₁ of polyethylene take the values 0, 1, 2or 3 (length unit), and a thickness L₂ of PVC take the values 0, 1, 2 or3 (length unit), for a given sample, the following values of thelikelihood function are obtained:

L₂ L₁ 0 1 2 3 0 4457 5173 5646 5925 1 5891 6068 6121 6075 2 6093 60445914 5721 3 5887 5705 5470 5192

We then have {circumflex over (L)}₁ ^(ech)=2 and {circumflex over (L)}₂^(ech)=1, the maximum value of the likelihood function being 6121.

Thus, according to the invention, to estimate the characteristicthicknesses, a plurality of energy spectra are used, associated with aplurality of combinations of known thicknesses of calibration materials.In the general case with M calibration materials, each calibrationmaterial y being able to take U_(y) different thicknesses, we have acalibration basis available comprising Π_(y=1) ^(M) U_(y) calibrationspectra. The number of possible thicknesses for each material can vary.Each calibration spectrum is noted S^(base)(L₁, L₂, . . . , L_(M)) withL₁, L₂, . . . , L_(M) the thicknesses of the materials MAT1, MAT2, . . ., MATM.

The spectrum transmitted through the sample to be characterised iscompared with those of the calibration basis by using the followinglikelihood function:

V(L ₁, L₂ , . . . L _(M))=Π_(j=1) ^(R) P(S _(j) ^(ech) |S _(j) ^(base)(L₁ , L ₂ , . . . L _(M)))   (4)

V(L₁, L₂, . . . L_(M)) describes the likelihood that the spectrumtransmitted through the sample corresponds to the spectrum transmittedthrough a stack of thicknesses L₁, L₂, . . . L_(M) of the M materials ofthe calibration basis.

S_(j) ^(ech) is the number of photons (or counts) counted in the channelj, in the spectrum transmitted through the sample, S^(ech) with R energychannels. The R components of the vector S^(ech) are independent randomvariables.

P(S_(j) ^(ech)|S_(j) ^(base) (L₁, L₂, . . . L_(M))) is the probabilitythat the channel j of the spectrum transmitted through the sample,corresponds to the channel j of the calibration spectrum associated withthicknesses L₁, L₂, . . . L_(M) of the materials 1 to M.

Thus, the likelihood function is equal to the product, for each of thechannels j, of the probabilities that the channel j of the spectrumtransmitted through the sample (measured spectrum) corresponds to thechannel j of the calibration spectrum associated with thicknesses L₁,L₂, . . . L_(M).

The function P describes a statistical modelling of a transmission ratethrough the sample, in each energy channel.

According to a particularly advantageous embodiment, the arrival ofphotons in each energy channel is assumed to follow a statisticalPoisson distribution. The choice of a Poisson distribution enables thebest estimation to be provided, this distribution best modelling thephysical reality in the spectrometer.

Thus, in each energy channel j, the probability to have S_(j) ^(ech)photons transmitted through the stack of the thicknesses L_(ki) of thematerials MATi, i=M, during a predetermined irradiation duration, isgiven by:

$\begin{matrix}{{P\left( S_{j}^{ech} \middle| v_{j} \right)} = {^{- v_{j}}\frac{v_{j}^{S_{j}^{ech}}}{S_{j}^{ech}!}}} & (5)\end{matrix}$

with v_(j) the number of photons transmitted through the sample in thechannel j, during an irradiation time T (identical for the spectrumtransmitted through the sample and for the calibration spectra).

-   -   P(S_(j) ^(ech)|v_(j)) is the probability of measuring S_(j)        ^(ech) counts for an expected value v_(i).

If it is assumed that the thicknesses of the materials 1 to M are thethicknesses L₁, L₂, . . . L_(M), there is:

v _(j) =μS _(j) ^(base)(L ₁ , L ₂ , . . . L _(M),)   (6)

μ corresponds to the drift of the spectrometer between the measurementof the calibration spectra and the measurement of the spectrumtransmitted through the sample. This drift is assumed to be zero,corresponding to μ=1, which is a quite realistic hypothesis.

The likelihood function is then expressed:

$\begin{matrix}\begin{matrix}{{V\left( {L_{1},L_{2},{\ldots \; L_{M}}} \right)} = {V\left( {S^{ech},{S^{base}\left( {L_{1},L_{2},{\ldots \; L_{M}}} \right)}} \right)}} \\{= {\underset{j - 1}{\overset{R}{\Pi}}{\exp \left( {- {S_{j}^{base}\left( {L_{1},L_{2},{\ldots \; L_{M}}} \right)}} \right)}\frac{{S_{j}^{base}\left( {L_{1},L_{2},{\ldots \; L_{M}}} \right)}^{S_{j}^{ech}}}{S_{j}^{ech}!}}}\end{matrix} & (7)\end{matrix}$

The characteristic thicknesses of the sample are obtained by searchingfor the maximum of the likelihood function. For convenience, it issimpler to try to maximise the logarithm of the likelihood function,which is quicker to calculate. There is in particular:

$\begin{matrix}{{\ln \left( {V\left( {L_{1},L_{2},{\ldots \; L_{M}}} \right)} \right)} \propto {{- {\sum\limits_{j = 1}^{R}\; {S_{j}^{base}\left( {L_{1},L_{2},{\ldots \; L_{M}}} \right)}}} + {\sum\limits_{j = 1}^{R}\; {S_{j}^{ech}\mspace{14mu} {\ln \left( {S_{j}^{base}\left( {L_{1},L_{2},{\ldots \; L_{M}}} \right)} \right)}}}}} & (8)\end{matrix}$

Then, the characteristic thicknesses of the M basis materials are givenby:

({circumflex over (L)} ₁ ^(ech) , {circumflex over (L)} ₂ ^(ech) , . . ., {circumflex over (L)} _(M) ^(ech))=argmax(ln(V(L ₁ , L ₂ , . . . L_(M))))   (9)

It can be seen that the invention implements a probabilistic approach,based on likelihood calculations and using as an estimation criterion,the maximum likelihood.

This approach is based on Bayes theorem, and on the hypothesis accordingto which all the combinations of thicknesses are equiprobable. Thishypothesis enables to establish that:

{circumflex over (θ)}=argmax(P(S ^(ech)|θ))   (10)

with S^(ech) the spectrum transmitted through the sample, θ=L₁, L₂, . .. L_(M), and {circumflex over (θ)}={circumflex over (L)}₁ ^(ech),{circumflex over (L)}₂ ^(ech), . . . , {circumflex over (L)}_(M) ^(ech)the estimated characteristic thicknesses.

Since the calibration basis only comprises a finite discrete number ofmaterial thicknesses, the likelihoods for the intermediate thicknessesare advantageously calculated by interpolation, either by interpolatingthe calibration spectra, or by interpolating the likelihood.

Interpolation enables the number of necessary calibration spectra to belimited. Linear or polynomial interpolation models can be used, or anyother interpolation model describing at best the behaviour of thespectra or of the likelihood as a function of the characteristicthicknesses of the calibration materials.

It can be an interpolation of calibration spectra. Thus, a larger numberof spectra named reference spectra is available, and these are used tocalculate the value of a likelihood function corresponding to thelikelihood of this reference spectrum taking into account the spectrumtransmitted through the sample S^(ech).

The reference spectra refer both to the calibration spectra and to theinterpolated spectra (values of an interpolation function of saidcalibration spectra).

In particular, an interpolation of the calibration spectra can beperformed by an interpolation function depending on the thickness of oneof the calibration materials. The calibration basis can thus be enhancedby calibration spectra named interpolated spectra. Subsequently, thevalues of a likelihood function are calculated from the spectrum of thesample S^(ech) and from the spectra of the enhanced calibration basis.The maximum of the values of the likelihood function is then searchedfor, this maximum being associated with the estimations of thecharacteristic thicknesses.

Preferably, an interpolation of the calibration spectra is performed byan interpolation function depending on at least one variable, eachvariable corresponding to the thickness of one of the calibrationmaterials. More preferentially, the interpolation function depends onseveral variables, corresponding to the thicknesses of each of thecalibration materials.

A linear interpolation can for example be performed on the logarithm ofthe energy spectra. Let a calibration basis with two calibrationmaterials. Let a thickness l₁ of the material MAT1 comprised between L₁^(k) and L₁ ^(k+1), and a thickness l₂ of the material MAT2 comprisedbetween L₂ ^(q) and L₂ ^(q+1). The spectrum transmitted through athickness l₁ of the material MAT1 joined to a thickness l₂ of thematerial MAT2 can be estimated by interpolating the spectra of thecalibration basis. For example the spectra in l₁,L₁ ^(q) (respectivelyin l₁,L₂ ^(q+1)) can be calculated from the spectra in L₁ ^(k), L₂ ^(q)and L₁ ^(k+1), L₂ ^(q) (respectively in L₁ ^(k), L₂ ^(q+1) and L₁^(k+1), L₂ ^(q+1)). For each channel j, the number of photons is givenby:

$\begin{matrix}{{\ln \left( {S_{j}^{base}\left( {l_{1},L_{2}^{var}} \right)} \right)} = {\ln\left( {{{S_{j}^{base}\left( {L_{1}^{k},L_{2}^{var}} \right)} + {{\ln \left( {S_{j}^{base}\left( {L_{1}^{k + 1},L_{2}^{var}} \right)} \right)}*\frac{L_{1}^{k} - l_{1}}{L_{1}^{k} - L_{1}^{k + 1}}{with}\mspace{14mu} L_{2}^{var}}} = {{L_{2}^{q}\mspace{14mu} {or}\mspace{14mu} L_{2}^{var}} = L_{2}^{q + 1}}} \right.}} & (11)\end{matrix}$

The spectrum in l₁, l₂ can then be interpolated from the previouslycalculated spectra in l₁, L₂ ^(q) and l₁, L₂ ^(q+1). The value of thespectrum in the channel j for a thickness l₁ of the material MAT1 joinedto a thickness l₂ of the material MAT2 is obtained as follows:

$\begin{matrix}{{\ln \left( {S_{j}^{base}\left( {l_{1},l_{2}} \right)} \right)} = {\ln\left( {{S_{j}^{base}\left( {l_{1},L_{2}^{q}} \right)} + {{\ln \left( {S_{j}^{base}\left( {l_{1},L_{2}^{q + 1}} \right)} \right)}*\frac{l_{1} - l_{2}}{l_{1} - L_{1}^{l + 1}}}} \right.}} & (12)\end{matrix}$

Alternatively, a non-linear interpolation is performed, for example by aLagrange polynomial or a cubic Hermite polynomial.

The interpolation by an interpolation function depending on thethickness of a calibration material enables a calibration spectrum to beobtained for each thickness of this calibration material located withina determined interval. Interpolation of spectra can also simply consistin increasing the number of available spectra, by adding to the initialcalibration spectra a finite number of spectra obtained byinterpolation.

Alternatively or additionally, an interpolation of the values of thelikelihood function is performed. Thus, a larger number of values isavailable, among which a maximum is searched for.

In particular an interpolation of the values of the likelihood functioncan be performed by an interpolation function depending on at least onevariable, each variable corresponding to the thickness of one of thecalibration materials. More preferentially the interpolation functiondepends on several variables, corresponding to the thicknesses of eachof the calibration materials.

Subsequently, the maximum of the values of the likelihood function issearched for, among the initial values and the values obtained byinterpolation. This maximum is associated with the estimations of thecharacteristic thicknesses.

For example, an interpolation of the values of the likelihood functionis performed using a function of two variables. Each variablecorresponds to one of the materials MAT1 or MAT2. This function isadvantageously non-linear, for example a second order polynomialfunction of the type:

F(L ₁ , L ₂)=a+bL ₁ +cL ₂ +dL ₁ L ₂ +eL ₁ ² +fL ₂ ²   (13)

where a, b, c, d, e, f are constants adjusted according to the leastsquare method, to the values calculated in step 32.

Interpolation by an interpolation function depending on the thickness ofa calibration material enables a value of the likelihood function to beobtained for each thickness of this calibration material located withina determined interval. Interpolation of the values of the likelihoodfunction can also simply consist in increasing a number of availablevalues of the likelihood function, by adding to the initial values afinite number of values obtained by interpolation.

Interpolations make it possible to resort to a reduced number ofcalibration spectra, while providing a high accuracy of estimation ofthe characteristic thicknesses.

Both interpolation types can be combined and/or implemented severaltimes to gradually refine the estimation of the characteristicthicknesses (iterative methods).

For example, an interpolation of the calibration spectra is performed toobtain a plurality of values of the likelihood function, then thesevalues are interpolated and a maximum of the obtained values is searchedfor after this second interpolation.

Several successive interpolations can also be implemented, reducing eachtime the interval of considered thicknesses and the pitch, as a functionof the previously obtained estimation. In particular, successive cyclesof searching a maximum of the values of the likelihood function and ofestimating the characteristic thicknesses are implemented. At eachcycle, a pitch of the thicknesses of calibration material is decreased,said thicknesses being associated with the calibration spectra and/orwith the values of the likelihood function. Preferably, the decrease inthis pitch comes along a decrease in a considered interval ofthicknesses. The calibration spectra here refer to the initialcalibration spectra and when necessary spectra obtained byinterpolation(s). The values of the likelihood function here refer tothe values calculated from the calibration spectra and when necessary tovalues directly obtained by interpolation(s).

The detector 403 is connected to a processor 404, configured toimplement the characterizing method according to the invention. Theprocessor 404 is connected to a memory 405 storing the calibrationspectra. The processor receives in input a spectrum transmitted throughthe sample, and outputs the estimations of the characteristicthicknesses of the sample.

FIG. 6 illustrates an advantageous exemplary method implementing twosuccessive cycles for searching a maximum of the values of thelikelihood function and estimating the characteristic thicknesses.

In a first step 61, the spectrum transmitted through the sample S^(ech)is acquired. Subsequently (step 62) the values of a likelihood functionare calculated from this spectrum S^(ech) and from each of thecalibration spectra S^(base)(L₁; L₂) of an initial calibration basis.The initial calibration basis corresponds to stacks of gauge blocks of athickness L₁ of the material MAT1 and a thickness L₂ of the materialMAT2, with:

-   -   L₁ extending over a first interval associated with the material        MAT1 (interval [0; L_(MAT1max)] for example) and according to a        first sampling pitch associated with the material MAT1

$\left( {{pitch}\frac{L_{{MAT}\; 1\mspace{14mu} \max}}{N\; 1}} \right),$

and

-   -   L₂ extending over a first interval associated with the material        MAT2 (interval [0; L_(MAT2max)] for example) and according to a        first sampling pitch associated with the material MAT2

$\left( {{pitch}\frac{L_{{MAT}\; 2\mspace{14mu} m\; {ax}}}{N\; 2}} \right).$

Thus N1*N2 spectra acquired with the combinations of thicknesses rangingfrom 0 to L_(MAT1 max) for the material MAT1 and from 0 to L_(MAT2 max)for the material MAT2 are obtained.

In step 65, an interpolation of the values of the likelihood function isperformed. Subsequently, the maximum of the values of the likelihoodfunction which are available after interpolation is searched for (step66). This maximum is associated with thicknesses {circumflex over (L)}′₁and {circumflex over (L)}^(′) ₂ respectively of the material MAT1 andthe material MAT2, and form approximate values of the estimations of thecharacteristic thicknesses.

In step 67, for each calibration material, a second respective intervalis determined, narrower than the first interval associated with the samecalibration material, and centred on the corresponding approximatevalue. The second intervals [L′₁₁; L′₁₂] and [L′₂₁; L′₂₂] are obtained.

Subsequently, an interpolation of the spectra of the initial calibrationbasis is performed on these second intervals [L′₁₁; L′₁₂] and [L′₂₁;L′₂₂] (step 68). An enhanced calibration basis is obtained, comprisingthe spectra S′^(base)(L₁; L₂).

The enhanced calibration basis corresponds to:

-   -   a thickness L₁ of the material MAT1 extending over the second        interval [L′₁₁; _(L′12)] (narrower than the interval [0;        L_(MAT1max)]) and according to a second respective sampling        pitch lower than the first sampling pitch associated with the        same calibration material (pitch lower than

$\left. \frac{L_{{MAT}\; 1\mspace{14mu} \max}}{N\; 1} \right),$

and

-   -   a thickness L₂ of the material MAT2 extending over the second        interval [L′₂₁; L′₂₂] (narrower than the interval [0;        L_(MAT2 max)]) and according to a second respective sampling        pitch lower than the first sampling pitch associated with the        same calibration material (pitch lower than

$\left. \frac{L_{{MAT}\; 2\mspace{14mu} m\; {ax}}}{N\; 2} \right).$

Step 62 of calculating the values of a likelihood function issubsequently reiterated, this time from the spectrum S^(ech) and fromeach of the spectra of the enhanced calibration basis S′^(base)(L₁; L₂).

Finally, the maximum of the thus calculated values is searched for (step69). This maximum is associated with thicknesses {circumflex over (L)}′₁^(ech) and {circumflex over (L)}₂ ^(ech) of the material MAT1 and thematerial MAT2, respectively, and form consolidated estimations of thecharacteristic thicknesses.

According to an alternative depicted in FIG. 7, a first performedinterpolation is an interpolation of the calibration spectra of theabove-defined initial calibration basis S^(base)(L₁; L₂) (step 78). Anenhanced calibration basis s″^(base)(L₁; L₂) is obtained, correspondingto the N1*N2 initial spectra, plus values obtained by interpolation. Theenhanced calibration basis is associated, for each calibration material,to a first respective interval of thicknesses.

Subsequently, the spectrum transmitted through the sample, S^(ech), isacquired (step 71), and the values of a likelihood function arecalculated from this spectrum S^(ech) and from each of the spectra ofthe enhanced calibration basis (step 72).

The maximum of the values of the likelihood function thus obtained issearched for (step 73). The thicknesses associated with this maximumform approximate values {circumflex over (L)}₁ ⁽²⁾ and {circumflex over(L)}₂ ⁽²⁾ of the estimations of the characteristic thicknesses.

Subsequently, in step 77, second intervals [L₁₁ ⁽²⁾, L₁₂ ⁽²⁾] and [L₂₁⁽²⁾; L₂₂ ⁽²⁾] are determined, each being associated with one of thecalibration materials. Each second interval is centred on the previouslyobtained approximate value associated with the same calibrationmaterial. Each second interval is narrower than the first interval ofthicknesses such as defined above, associated with the same calibrationmaterial and with the enhanced calibration basis.

Then, an interpolation of the values of the likelihood function isperformed (step 75). Subsequently, the maximum of the values of thelikelihood function is performed (new iteration of step 73), this timeamong the values of the likelihood function available after this secondinterpolation. This maximum is associated with thicknesses {circumflexover (L)}₁ ^(ech) and {circumflex over (L)}₂ ^(ech) respectively of thematerial MAT1 and the material MAT2, and form consolidated estimationsof the characteristic thicknesses.

As specified above, the calibration spectra can be experimentallyobtained, using stacks of gauge blocks, each gauge block being made of acalibration material and having a known respective thickness, ornumerically simulated. Thus, a first calibration basis is obtained.

Each calibration material has an effective atomic number. The maximumand the minimum of the effective atomic numbers of the consideredcalibration materials define together an interval of effective atomicnumber. If the mean effective atomic number of the sample is outsidethis interval, at least one of the associated characteristic thicknessescan be negative. For example, if the material MAT1 is polyethylene(Z_(eff)=5.53), the material MAT2 is PVC (Z_(eff)=4.23), a sample ofiron (Z_(eff)=26), of chromium (Z_(eff)=24), or of chlorine(Z_(eff)=17), will be characterised by a negative characteristicthickness of polyethylene. We are therefore in an area not covered bythe first calibration basis.

A common solution consists in performing an extrapolation of the pointsof the first calibration basis.

A more reliable solution is here provided. The aim is to enhance thefirst calibration basis using measurements or simulations implying anadditional standard (or additional gauge block) of a determinedthickness made of a reference material, distinct from the calibrationmaterials MAT1 and MAT2. Said reference material has an effective atomicnumber outside the above-described interval. The characteristicthicknesses of this additional standard, in the calibration materialbasis, are known, and at least one of these characteristic thicknessesis negative. In other words, the spectrum of the energy transmittedthrough the additional standard is equal to the spectrum of the energytransmitted through a virtual stack of gauge blocks of a calibrationmaterial, one gauge block at least having a negative thickness (hencethe term “virtual”).

The first calibration basis is thus enhanced by points associated withnegative characteristic thicknesses. The characteristic thicknessesassociated with said additional standard can be determined using amethod according to prior art such as described in the introduction.

Thus, each calibration spectrum corresponds to the spectrum of theenergy transmitted through a stack of gauge blocks each formed of aknown thickness of a calibration material, and when necessary a gaugeblock having a negative thickness (virtual gauge block).

In FIG. 8, the points of a thus enhanced calibration basis areschematically represented. A first series of points corresponds tocombinations of polyethylene and PVC, the thicknesses of polyethyleneand of PVC taking integer values between 0 and 10 included (lengthunit). This calibration basis is enhanced by measurements of energyspectra of gauge blocks of different thicknesses of calcium (Z_(eff)=20,points 701), of sulphur (Z_(eff)=16, points 702), of beryllium(Z_(eff)=4, points 703).

The invention is not limited to a decomposition in a basis of twocalibration materials. A decomposition in a basis of more than twocalibration materials, for example three, four or more can be performed.Furthermore, the considered calibration materials do not have to meetany particular condition. The use of bases with more than 2 materialscan prove to be useful when there is a continuity of the energy spectrum(known as “k-edge”) in the measured energy range.

The method according to the invention can comprise, after estimation ofthe characteristic thicknesses, an additional step of processing,comprising for example the estimation of a concentration in the sample,or an estimation of the effective atomic number of the sample.

In particular, a function ƒ can be determined, such as:

Z _(eff)=ƒ(ρ₁ {circumflex over (L)} ₁ ^(ech), ρ₂ {circumflex over (L)} ₂^(ech), . . . , ρ_(M) {circumflex over (L)} _(M) ^(ech))   (14)

with M being the calibration materials, ρ_(i) the density of thecalibration material i, {circumflex over (L)}_(i) ^(ech) the estimationof the characteristic length associated with the calibration material i.

There is for example:

$\begin{matrix}{{{f(x)} = {a + {bx} + {cx}^{2} + {dx}^{3}}},{{{with}\mspace{14mu} x} = \frac{\rho_{1}{\hat{L}}_{1}^{ech}}{{\rho_{1}{\hat{L}}_{1}^{ech}} + {\rho_{2}{\hat{L}}_{2}^{ech}}}}} & (15)\end{matrix}$

with a, b, c, d real numbers and for example polyethylene for MAT1 andPVC for MAT2.

If the basis contains more than two materials, the following functioncan be used:

$\begin{matrix}{{f\left( {\rho_{i},{\hat{L}}_{i}^{ech}} \right)} = \left( \frac{\sum_{i = 1}^{M}{\rho_{i}{\hat{L}}_{i}^{ech}Z_{eff}^{p}}}{\sum_{i = 1}^{M}{\rho_{i}{\hat{L}}_{i}^{ech}}} \right)^{1\text{/}p}} & (16)\end{matrix}$

p being estimated from measurements performed on materials of knownZ_(eff).

The invention is especially applied in the medical field, especially foranalysing biological samples by spectral tomography.

The invention is not limited to the examples which have just beendeveloped, and numerous alternatives can be imagined without departingfrom the scope of the present invention, for example other calibrationmaterials, other types of interpolations, etc. For example, otherdefinitions of the likelihood function can be considered, based on otherstatistical modelling of the transmission rate through the sample in thespectrometer.

1. A method for characterizing a sample, by estimating a plurality ofthicknesses, named characteristic thicknesses, each being associatedwith a material named calibration material, wherein said methodcomprises the following steps: acquiring an energy spectrum, namedspectrum transmitted through the sample (S^(ech)), said spectrum beingdefined by a number of photons transmitted through the sample in eachchannel of a plurality of energy channels located in an X and/or gammaspectral band; for each spectrum of a plurality of energy spectra namedcalibration spectra (S^(base)(L₁; L₂)), calculating the value of alikelihood function from said calibration spectrum (S^(base)(L₁; L₂))and from the spectrum transmitted through the sample (S^(ech)), eachcalibration spectrum (S^(base)(L₁; L₂)) corresponding to the spectrumtransmitted through a stack of gauge blocks, each gauge block beingformed of a known thickness of a calibration material; determining theestimations of the characteristic thicknesses ({circumflex over (L)}₁^(ech); {circumflex over (L)}₂ ^(ech)) associated with the sample, fromsaid values of a likelihood function and according to the criterion ofmaximum likelihood.
 2. The method according to claim 1, whereindetermining the estimations of the characteristic thicknesses (L₁^(ech); L₂ ^(ech)) associated with the sample, comprises searching for amaximum of the values of the likelihood function, the thicknessesassociated with this maximum forming the estimations ({circumflex over(L)}₁ ^(ech); {circumflex over (L)}₂ ^(ech)) of the characteristicthicknesses (L₁ ^(ech); L₂ ^(ech)).
 3. The method according to claim 1,wherein said method comprises at least one step of interpolating thevalues of the likelihood function or of interpolating the calibrationspectra.
 4. The method according to claim 3, wherein at least one stepof interpolating implements a non-linear interpolation function.
 5. Themethod according to claim 3, wherein said method comprises the followingsteps: interpolating values of the likelihood function by a likelihoodinterpolation function depending on at least one variable, each variablecorresponding to the thickness of a calibrating material; and searchingfor a maximum of the values of said likelihood interpolation function,the thicknesses associated with this maximum forming the estimations ofthe characteristic thicknesses.
 6. The method according to claim 3,wherein it comprises the following steps: Interpolating the calibrationspectra by a spectrum interpolation function depending on at least onevariable, each variable corresponding to the thickness of a calibrationmaterial; and searching for a maximum of the values of said spectruminterpolation function, the thicknesses associated with this maximumforming the estimations of the characteristic thicknesses.
 7. The methodaccording to claim 3, wherein the following steps are implemented:interpolating the values of the likelihood function by a likelihoodinterpolation function, said values being associated with combinationsof known thicknesses of calibration materials such that for eachcalibration material, the associated thicknesses are located within afirst respective interval; searching for a maximum of the values of saidlikelihood interpolation function, the thicknesses associated with thismaximum forming approximate values ({circumflex over (L)}′₁, {circumflexover (L)}′₂) of the estimations of the characteristic thicknesses;interpolating the calibration spectra by a spectrum interpolationfunction depending on at least one variable, each variable correspondingto the thickness of a calibration material and taking values locatedwithin a second respective interval ([L′₁₁, L′₁₂]; [L′₂₁, L′₂₂])narrower than the first interval associated with the same calibrationmaterial and centred on the approximate value ({circumflex over (L)}′₁,{circumflex over (L)}₂) associated with the same calibration material;for each of the values of said spectrum interpolation function,calculating the value of the likelihood function and searching for amaximum of said values of the likelihood function, the thicknessesassociated with this maximum forming consolidated estimations({circumflex over (L)}₁ ^(ech); {circumflex over (L)}₂ ^(ech)) of thecharacteristic thicknesses.
 8. The method according to claim 3, whereinthe following steps are implemented: interpolating calibration spectraby a spectrum interpolation function, said calibration spectra beingassociated with combinations of known thicknesses of calibrationmaterials such that for each calibration material, the associatedthicknesses are located within a first respective interval; for each ofthe values of said spectrum interpolation function, calculating thevalue of the likelihood function and searching for a maximum of saidvalues of the likelihood function, the thicknesses associated with thismaximum forming approximate values ({circumflex over (L)}₁ ⁽²⁾,{circumflex over (L)}₂ ⁽²⁾) of the estimations of the characteristicthicknesses; interpolating the values of the likelihood function by alikelihood interpolation function depending on at least one variable,each variable corresponding to the thickness of a calibration materialand taking values located within a second respective interval ([L₁₁ ⁽²⁾;L₁₂ ⁽²⁾]; [L₂₁ ⁽²⁾; L₂₂ ⁽²⁾]) narrower than the first intervalassociated with the same calibration material and centred on theapproximate value ({circumflex over (L)}₁ ⁽²⁾, {circumflex over (L)}₂⁽²⁾) associated with the same calibration material; and searching for amaximum of the values of said likelihood interpolation function, thethicknesses associated with this maximum forming consolidatedestimations ({circumflex over (L)}₁ ^(ech); {circumflex over (L)}₂^(ech)) of the characteristic thicknesses.
 9. The method according toclaim 1, wherein the likelihood function is determined from astatistical modelling of the spectrum transmitted through the sample(S^(ech)), according to Poisson distribution.
 10. The method accordingto claim 1, wherein the likelihood function calculated from saidcalibration spectrum and from the spectrum transmitted through thesample, is defined by:${\ln \left( {V\left( {S^{ech},{S^{base}\left( {L_{1},\ldots,L_{M}} \right)}} \right)} \right)} = {C\left( {{- {\sum\limits_{j = 1}^{R}\; {S_{j}^{base}\left( {L_{1},\ldots,L_{M}} \right)}}} + {\sum\limits_{j = 1}^{R}\; {S_{j}^{ech}{\ln \left( {S_{j}^{base}\left( {L_{1},\ldots,L_{m}} \right)} \right)}}}} \right)}$V is the likelihood function, ln is the naperian logarithm,S^(ech)=Σ_(j=1) ^(R)S_(j) ^(ech) is the spectrum transmitted through thesample, presenting j=R energy channels, S^(base)(L₁, . . . ,L_(M))=Σ_(j=1) ^(R)S_(j) ^(base)(L₁, . . . , L_(M)) is the calibrationspectrum with j=R energy channels of the combination of M calibrationmaterials each associated with a respective thickness L₁, . . . , L_(M),C is a constant.
 11. The method according to claim 1, wherein at leastone calibration spectrum (S^(base)(L₁; L₂)) corresponding to thespectrum transmitted through an additional standard is used, thisadditional standard consisting of a determined thickness of a referencematerial, and being associated with a combination of known thicknessesof calibration materials such that at least one thickness takes anegative value.
 12. The method according to claim 1, wherein a step ofmanufacturing a calibration basis comprising said calibration spectra(S^(base)(L₁; L₂)) is carried out, comprising: measurements of thespectra transmitted through each of a plurality of stacks of gaugeblocks, each gauge block consisting of a known thickness of acalibration material; a measurement of the spectrum transmitted throughat least one additional standard consisting of a determining thicknessof a reference material, and associated with a combination of knownthicknesses of the calibration materials such that at least onethickness takes a negative value; ranking of all the measured spectrainto a single database connecting a spectrum to a combination ofthicknesses of the calibration materials.
 13. The method according toclaim 1, comprising calculating the mean effective atomic number of thesample, as a function of the estimations of the characteristicthicknesses (L₁ ^(ech); L₂ ^(ech)).
 14. A computer program productarranged to implement a method according to claim
 1. 15. A device forcharacterizing a sample, comprising an electromagnetic source emittinginto an X and/or gamma spectral band, and a detector for measuring aspectrum (S^(ech)) transmitted through the sample, said spectrum(S^(ech)) being defined by a number of photons transmitted through thesample in each channel of a plurality of energy channels, wherein itcomprises a processor configured to implement the method according toclaim 1, and a memory receiving the calibration spectra (S^(base)(L₁;L₂)), and connected to the processor.